Lotka's distribution and distribution of co-author pairs’ frequencies

Abstract

The original Lotka's Law refers to single scientist distribution, i.e. the frequency of authors A_i with i publications per author is a function of i: A_i = f(i). However, with increasing collaboration in science and in technology the study of the frequency of pairs or triples of co-authors is highly relevant. Starting with pair distribution well-ordered collaboration structures of co-author pairs will be presented, i.e. the frequency of co-author pairs N_ij between authors with i publications per author and authors with j publications per author is a function of i and j: N_ij = f(i, j) using the normal count procedure for counting i or j. We have assumed that the distribution of co-author pairs’ frequencies can be considered to be reflection of a social Gestalt and therefore can be described by the corresponding mathematical function based on well-known general characteristics of structures in interpersonal relations in social networks. We have shown that this model of social Gestalts can better explain the distribution of co-author pairs than by a simple bivariate function in analogy to Lotka's Law. This model is based on both the Gestalt theory and the old Chinese Yin/Yang theory.

Introduction

Since several decades, collaboration is increasing in science and in technology. Usually, the bibliometric method for the study of collaboration is the investigation of co-authorships (de B. Beaver, 2001; de Solla Price, 1963; Glänzel, 2002; Glänzel & de Lange, 1997; Glänzel & Schubert, 2004; Luukkonen, Persson, & Silvertse, 1992; Miquel & Okubo, 1994; Newman, 2001; Okubo, Miquel, Frigoletto, & Doré, 1992; Tijssen & Moed, 1989; Zitt, Bassecoulard, & Okubo, 2000).

The original Lotka's Law (Lotka, 1926) refers to single scientist distribution, i.e. the frequency of authors A_i with i publications per author is a function of i:

$$A_i=f(i)$$

However, with increasing collaboration in science and in technology the study of the frequency of pairs or triples of co-authors is highly relevant. Starting with pair distribution the frequency of co-author pairs N_ij between authors with i publications per author and authors with j publications per author is a function of i and j:

$$N_{ij}=f(i,j)$$

Whereas regarding Lotka's Law single scientists P distribution (both in single authored and in multi-authored bibliographies) is of interest, in the future pairs P, Q distribution, triples P, Q, R distribution, etc., should be considered.

Starting with pair distribution, the following questions arise in the present paper:

• Is there any regularity for the distribution of co-author pairs’ frequencies?

• If yes, can the distribution of co-author pairs be described by an extension of Lotka's Law or can this distribution be better described by a model of social Gestalts?

Regarding the last question, two theoretical distributions will be calculated and these distributions will be further specified and discussed in the next sections.

However, in Section 1 we only intend to visualize the two theoretical distributions in comparison with an empirical distribution.

For visualization, Lotka's distribution and co-authorship distribution of pairs of collaborators obtained from the journal Science are presented in Fig. 1. The articles from 1980 to 1998 of the journal Science were studied with 47,117 authors and the total sum of N_ij = 418,458 co-author pairs (method for counting N_ij, cf. below).

Fig. 1 shows the distribution of the number of authors A_i with i publications per author (upper row) contrasted with the distribution of the number of co-author pairs N_ij between authors with i publications per author and authors with j publications per author (lower row). For clarity's sake and optimum visualization the presentation of data are restricted to authors with at most 10 articles i (i = 1, 2, …, 10) or j (j = 1, 2, …, 10) respectively.

The upper row reflects the Lotka's Law. In this row, the distribution of the number of authors A_i with i publications per author is given on the left, and on the right the corresponding double logarithmic presentation. A_i (or log A_i respectively) is plotted at the Y-axis and i (or log i respectively) is plotted at the X-axis.

In the lower row, the distribution of the number of co-author pairs N_ij is given on the left, and the corresponding triple logarithmic presentation on the right. The number of pairs N_ij (or log N_ij respectively) is plotted at the Z-axis, i (or log i respectively) is plotted at the X-axis and j (or log j respectively) is plotted at the Y-axis.

Fig. 1 has shown we can say yes to the first question that there is any regularity existing for the distribution of co-author pairs’ frequencies.

However, before considering the second question regarding the description of the distribution of co-author pairs’ frequencies [N_ij = f(i, j)] information about the method of counting these co-author pairs is necessary.

Method for counting N_i and N_ij (in extended form, cf. Appendix A): Given is an artificial bibliography including eight papers (names of authors: A, B, …).

The number of publications i (or j respectively) per author P (or Q respectively) is determined by resorting to the “normal count procedure”. Each time the name of an author appears, it is counted (e.g. A three times: once in the first paper, and once each in the 4th and 8th papers).

Pairs P, Q are marked in the cells of the matrix under the condition of both the first authors P count (i) and the second authors Q count (j), i.e. the authors are ordered according to i or j respectively in both the row and the column (cf. Table 1).

Under the condition, the place of the authors in the by-line is not taken into consideration the symmetrical matrix is resulting. For example, the pair G, A is marked two times: once under the condition G count (i) and A count (j) and once under the condition A count (i) and G count (j).

In the symmetrical matrix, one can determine for each author P the number of his collaborators N_P. N_P is equal to the Degree Centrality in Social Network Analysis (SNA).

The matrix of N_ij (Table 2, derived from the symmetrical matrix) is the representation of the number of pairs N_ij with authors who have i publications per author, with authors who have j publications per author included in the bibliography.

For example, the pairs E, D and F, D in Table 1 are counted both as N₁₂ = 2 and N₂₁ = 2 in the matrix of N_ij.

As mentioned above in Section 1 we only intend to visualize both the theoretical pattern derived from the extension of Lotka's Law and the theoretical pattern of social Gestalt in comparison with the empirical distribution obtained from the journal Science.

Fig. 2 shows the comparison of the empirical distribution of log N_ij with the two different distributions of theoretical values. The figures on the right are rotated by 90°.

The empirical distribution of the pairs’ frequencies in Science (Fig. 2, second row) is rather equal to the theoretical distribution of social Gestalts (third row) but different from the other, i.e. from the pattern by the extension of Lotka's Law (first row).

This is the first proof for the assumption the distribution of the co-author pairs’ frequencies N_ij can be considered to be a social Gestalt. It is the key phrase of this paper stating a key result and it constitutes the principal method and the main point of the paper.

Both the extension of Lotka's Law and the meaning of Gestalt will be explained in the next sections followed by the mathematical function for the description of social Gestalts. This general theoretical function is valid for different kinds of social Gestalts. We will give the proof that the distribution of co-author pairs’ frequencies is one example of them. Four journals are studied.

For thorough explanation of the theoretical and methodological background of the studies in this paper, in most of the sections or paragraphs reference is given to the corresponding information in the annex.